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The Hermitian R‐Conjugate Generalized Procrustes Problem

Author

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  • Hai-Xia Chang
  • Xue-Feng Duan
  • Qing-Wen Wang

Abstract

We consider the Hermitian R‐conjugate generalized Procrustes problem to find Hermitian R‐conjugate matrix X such that ∑k=1p∥AkX-Ck∥2 + ∑l=1q∥XBl-Dl∥2 is minimum, where Ak, Ck, Bl, and Dl (k = 1,2, …, p, l = 1, …, q) are given complex matrices, and p and q are positive integers. The expression of the solution to Hermitian R‐conjugate generalized Procrustes problem is derived. And the optimal approximation solution in the solution set for Hermitian R‐conjugate generalized Procrustes problem to a given matrix is also obtained. Furthermore, we establish necessary and sufficient conditions for the existence and the formula for Hermitian R‐conjugate solution to the linear system of complex matrix equations A1X = C1, A2X = C2, …, ApX = Cp, XB1 = D1, …, XBq = Dq (p and q are positive integers). The representation of the corresponding optimal approximation problem is presented. Finally, an algorithm for solving two problems above is proposed, and the numerical examples show its feasibility.

Suggested Citation

  • Hai-Xia Chang & Xue-Feng Duan & Qing-Wen Wang, 2013. "The Hermitian R‐Conjugate Generalized Procrustes Problem," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).
    • Handle: RePEc:wly:jnlaaa:v:2013:y:2013:i:1:n:423605
    DOI: 10.1155/2013/423605
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    References listed on IDEAS

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    1. Chang-Zhou Dong & Qing-Wen Wang & Yu-Ping Zhang, 2012. "On the Hermitian R‐Conjugate Solution of a System of Matrix Equations," Journal of Applied Mathematics, John Wiley & Sons, vol. 2012(1).
    2. J. Gower, 1975. "Generalized procrustes analysis," Psychometrika, Springer;The Psychometric Society, vol. 40(1), pages 33-51, March.
    3. Bert Green, 1952. "The orthogonal approximation of an oblique structure in factor analysis," Psychometrika, Springer;The Psychometric Society, vol. 17(4), pages 429-440, December.
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